lecture on safety and security of information systems lesson 6 7548

ASYMMETRIC CIPHERS


Contents
1)

Principles Of Public-Key Cryptosystems

2)

RSA Algorithm


1. Principles Of Public-Key Cryptosystems


1. Principles Of Public-Key Cryptosystems
 Commonly know as public key cryptography
 Invented by Whitfield Diffie and Martin Hellman in 1976
 Uses a pair of key
 A private key that is kept secret
 A public key that can be sent to anyone


Public-Key Cryptosystems
 Asymmetric algorithms rely on one key for encryption and a different but

related key for decryption. These algorithms have the following important
characteristic.
 It is computationally infeasible to determine the decryption key given only

knowledge of the cryptographic algorithm and the encryption key.
 Either of the two related keys can be used for encryption, with the other used

for decryption.


Encryption with public key


Encryption with private key


Authentication and confidentiality
 possible to provide both the authentication function and confidentiality by

a double use of the public-key.
 Z=E(PUb,E(PRa,X))
 X=D(PUa,D(PRb,Z))


Applications for Public-Key Cryptosystems
 Encryption/decryption: The sender encrypts a message with the

recipient’s public key.
 Digital signature: The sender “signs” a message with its private key.
 Key exchange: Two sides cooperate to exchange a session key.


Requirements for Public-Key Cryptography
 It is computationally easy for a party B to generate a pair.

 It is computationally easy for a sender A, knowing the public key and the

message to be encrypted,M, to generate the corresponding ciphertext.
C=E(PUb,M)
 It is computationally easy for the receiver B to decrypt the resulting

ciphertext using the private key to recover the original message:


Requirements for Public-Key Cryptography
 It is computationally infeasible for an adversary, knowing the public

key,PUb,to determine the private key,PRb.
 It is computationally infeasible for an adversary, knowing the public key,

PUb, and a ciphertext, C, to recover the original message, M.


2. RSA ALGORITHM


RSA Algorithm
 Developed in 1977 by Ron Rivest, Adi Shamir, and Len Adleman.
 The RSA scheme is a block cipher in which the plaintext and ciphertext

are integers between 0 and n-1 for some n. A typical size for n is 1024 bits,

or 309 decimal digits. That is, n is less than 21024
 Based on exponentiation in a finite field over intergers modulo a prime

Description of the Algorithm
 Select two large prime numbers: p and q
 Calculate: n = pq
 Calculate: m=(p-1)(q-1)
 Choose a small number e, co prime to m, with GCD(m,e)=1; 1 Find d, such that e.d ≡ 1 mod m
 PU = (n, e), PR = (n, d)


Description of the Algorithm
 Encryption:

C = Me mod n (với M < n)
 Decryption:

M = Cd mod N


Euclid’s algorithm
 Computing the greatest common divisor (GCD) of two numbers,

gcd(a,b) = gcd(b, a mod b)

1. A ← a; B ← b
2. if B = 0 return A = gcd(a, b)
3. R = A mod B

4. A ← B
5. B ← R

6. goto 2



Extended Euclid’s algorithm
1.

(A1, A2, A3) ← (1, 0, m); (B1, B2, B3) ← (0, 1, b)

2.

if B3 = 0 return A3 = gcd(m, b); no inverse

3.

if B3 = 1 return B3 = gcd(m, b); B2

4.

Q = A3 div B3

5.

(T1, T2, T3) ←(A1 – Q*B1, A2 – Q*B2, A3 – Q*B3)

6.

(A1, A2, A3) ← (B1, B2, B3)

7.

(B1, B2, B3) ← (T1, T2, T3)

8.

goto 2


Extended Euclid’s algorithm - example
 Finding inverse of 7 in modulo 187

=>Result: 80


RSA Example
 p = 11, q = 3 => n = pq=33
 m= (p-1)(q-1) = (11 – 1)(3 – 1) = 20
 Gcd(m,e)=1
 e corprime to m, means that the largest numbet that can be exactly divide

both e and m (their greatest common divisor, or gcd) is 1. Euclid's

algorithm is used to find the GCD of two numbers


RSA Example
 e=2 => GCD(20,e) = 2 (no)
 e=3 => GCD(20,e)=1 (yes!)
 Find d: using Extended Euclid's algorithm ? d=7

 PU (33, 3), PR = (33, 7)

Plaintext: M = 15:
Encryption: C = 153mod 33 = 9

Deencryption: c=9
M = 97mod 33 = 15


RSA Security
 Brute-force attack
 Mathematical attack
 Timing attack
 Chosen ciphertext attack